From the minimality of the realizations in (REF ), we have that the pair \((A_1,B_1)\)  is implicitly stabilizable and that the finite zeros of \(\mathbf {G}_2\)  are the finite generalized eigenvalues (see chapter 12 of [1]} for more details) of the pencil \(\mathbf {S}(\lambda )\overset{def}{=}\footnotesize \begin{bmatrix}A_2-\lambda I_{n_2}&B_2\\C_2&D_2\end{bmatrix}\) . Yet, since \(\mathbf {G}_2\)  has full row normal rank along with no zeros in \(\mathbb {C}\backslash \mathbb {S}\) , it follows by Lemma 3.33 and Theorem 3.34 in [2]} that \(\mathbf {S}(\lambda )\)  has full row rank \(\forall \lambda \in \mathbb {C}\backslash \mathbb {S}\) . This further implies that the following pencil
\(\small \begin{bmatrix}A_1-\lambda I_{n_1}&B_1C_2&B_1D_2\\O&A_2-\lambda I_{n_2}&B_2\end{bmatrix}=\\=\small \begin{bmatrix}A_1-\lambda I_{n_1}&B_1&O\\O&O&I_{n_2}\end{bmatrix}\footnotesize \begin{bmatrix}I_{n_1}&O&O\\O&C_2&D_2\\O&A_2-\lambda I_{n_2}&B_2\end{bmatrix}\) 
